How to recognize

152

G. Helmberg

Institut für Technische Mathematik und Geometrie

Universität Innsbruck, Technikerstrasse 13, A-6020 Innsbruck Gilbert.Helmberg@uibk.ac.at

How to recognize

functions in L _p ( ^R ) + L _q ( ^R )

Consider two function spaces Lp = Lp(^R)and Lq =Lq(^R) (0<

p≤q ≤^∞). The interest in the space Lp+Lq = {f = fp+fq : fp∈Lp, fq∈Lq}originates in Fourier analysis [2 (p.18)]: for any function f = f₁+f₂ ∈ L₁+L₂it is possible to define a Fourier transform ˆf= ˆf₁+ ˆf₂∈C0+L2and ˆf is well-defined even if the representation of f= f₁+f₂is not unique.

This definition extends the Fourier transform to all functions f ∈ Ls(1≤ s≤ 2), since it is easy to see that Ls ⊂ Lp+Lqfor p <

s<q [1 (13.19)]; for any f ∈Lswe have Z

{| f |>1}

|f|^pdx≤ Z {| f |>1}

|f|^sdx<∞, Z

{| f |≤1}

|f|^qdx≤ Z {| f |≤1}

|f|^sdx<∞ if q<∞,

f =f 1{| f |>1}+f 1_{{| f |≤1}}∈Lp+Lq. (1)

Not every function in Lp+Lq, however, needs to belong to some space Ls, as demonstrated by the functions f ∈ L₁+L₂defined by f(x) =x^αfor α∈] −1,−¹₂[.

If one wants to apply a Fourier transformation ˆf= ˆf₁+ ˆf₂ to a given function f on R, one has to make sure that f belongs to L1+L2and one has to exhibit the components f1and f2of some representation of f as in (1). Since in general|fp+fq| may be small if|fp|and|fq| are both large and either of these may be small if|fp+fq|is large it is not obvious that in general the func- tions f^> = f 1{| f |>1} and f^<= f 1_{{| f |≤1}}

serve to determine indices p and q and furnish a decomposition as in (1). Concerning the latter remark we have the following the- orem.

Theorem. A complex-valued function f belongs to L_p+L_q(0< _p≤ q≤^∞)if and only if f^>∈Lpand f^<∈Lq.

Proof. Since f = f 1{| f |>1}+f 1_{{| f |≤1}}the ‘if’-part is clear.

Conversely, if f = fp+fq( fp∈Lp, fq∈Lqwithout loss of gener- ality we assume 0<_p≤q≤^∞), then we have

|f|1{| f |>1}≤ |f|1_{{| f}

p|>¹₂}+|f|1_{{| f}

q|>¹₂}

≤ |fp|1_{{| f}

p|>¹₂}+|fq|1_{{| f}

p|>¹₂}+|fp|1_{{| f}

q|>¹₂}+|fq|1_{{| f}

q|>¹₂}. (2) Since the sets{|fp| > ¹₂}and{|fq| > ¹₂}have finite measure, all four functions on the right side of (2) and therefore also f 1{| f |>1}

belong to Lp. Furthermore,

|f|1_{{| f |≤1}}≤ (|fp|+|fq|)1_{{| fp|≤1,| fq|≤1}}+1_{{| fp|>1}}+1_{{| fq|>1}}

≤ |f_p|1_{{| fp|≤1}}+|f_q|1_{{| fq|≤1}}+1_{{| fp|>1}}+1_{{| fq|>1}}. (3) Again all four functions on the right side of (3) belong to Lq, there-

fore also f 1_{{| f |≤1}}. 

Since for a given function f on R the integrals^R{| f |>1}|f|^sdx and R

{| f |≤1}|f|^sdx are monotone increasing respectively decreasing functions of s we obtain Lp+Lq ⊂ L_p^′+L_q^′ for 0 < _p^′ ≤ p ≤ q≤q^′≤^∞. For a given function f on R having the property that

f^> ∈Lpand f^<∈Lq(0<_p≤q≤^∞) define

p =sup{p : f^>∈Lp}, q=inf{q>_{0 : f<} ∈Lq}. Then, for finite p respectively q, the integrals^R{| f |>1}|f|^pdx and R

{| f |≤1}|f|^qdx may be finite or not [1 (13.28)].

As a consequence of the theorem we obtain the following corol- lary:

Corollary. If p > _{q then f} ∈ L_s for all s ∈ ]q, p[. If p ≤ q then f ∈ Lp+Lqfor all p<p and all q>_{q. If p}<_{q then f /}∈ Lsfor all s>_0.

The mentioned statements can be carried over to functions on a σ-finite, infinite non-atomic measure space. k

References

1 Hewitt, Edwin/Stromberg, Karl: Real and Abstract Analysis. Springer Verlag Berlin Heidelberg New York, 1965.

2 Stein, Elias M./Weiss, Guido: Introduc- tion to Fourier analysis on Euclidean spaces.

Princeton University Press, 1971.